TESTS OF SIGNIFICANCE IN A 2×2 CONTINGENCY TABLE EXTENSION OF FINNEY'S TABLE: EXTENSION OF FINNEY'S TABLE

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TESTS OF SIGNIFICANCE IN A 2 x 2 CONTINGENCY TABLE:

EXTENSION OF FORNEY'S TABLE

COMPUTED BY

R. LATSCHA

Institute of Actuarial Science, University of Berne

EDITORIAL FOBBWOBD

Finney (1948) has given a table which may be used to test the significance of the deviation

from proportionality in any 2 x 2 contingency table having both the frequencies in one of

its margins less than or equal to 16. The table printed below extends the range of Finney's

table up to marginal frequencies of 20. As the interpretation and uses of the new table are

exactly similar to those of the 1948 table, only a brief introductory statement is required.*

Using Finney's notation, the contingency table should be arranged in the form

Series I

Series II

Number of

Successes

a

b

r = a + b

Failures

A-a

B-b

A+B-a-b

Total

A

B

A+B

where series I is defined to be that which makes A^B, and the type of observation

con-ventionally regarded as a ' success' is that which makes a/A > b/B. The table of significance

levels is arranged in sections according to the value of A; the sections for A = 9(1) 15 were

given by Finney, while those for A = 16(1)20 computed by Latscha are printed below.

For given data, the table is entered in the section for A, the subsection for B and the

line for a; then the main body of the table shows in bold type the appropriate significance

points for b. Thus if the observed value of 6 is equal to or less than the bold integer in the

column headed 0-05, 0-025, 0-01 or 0-005, then a/A is significantly greater than b/B

(single-tail test) at these probability levels. On the other hand, for the two-(single-tail test, if 6 is equal

to or less than the integer in a given column, a/A is significantly different from b/B at

a probability level equal to twice the figure heading that column, i.e. at the 0-10, 0-05,

0-02 and 0-01 levels, respectively. A dash, or absence of an entry, for some combination of

A, B and a indicates that no 2 x 2 table in that class can show a significant effect at that

level.

Owing to the discontinuous character of the hypergeometric distribution, the

con-ditional probability that, for a given value of o + 6, b will be equal to or less than the value

specified in bold type will generally be leas, and often very considerably less, than that

shown at the head of the column; the true probabilities are given in small type.

* Copies of Finney's table are available as a Biomebrika 'separate' and the present extension will

be made available in similar form. Finney's table, but not the extension, is included in the new

R. LATSOHA

75

As an illustration, we may use Lange'e data on criminality among twin brothers or

sisters of criminals (Fisher, 1946, §21-01). This example was taken by Finney (1948), but

as A > 15 he used it to show how his table could be extended under certain conditions. As

A < 20, direct entry is now possible in Latecha's table.

The contingency table shows the number of twin brother or sisters of criminal who had

also been convicted of crime, classing separately monozygotic and dizygotic (but

like-sexed) twins:

Dizygotic

Monozygotic

Total

Not convicted

15 ( = o)

3 (=6)

18

Convicted

2

10

12

Total

17 (=A)

13 (=B)

30

Following the rule given above, the letters A, B, a and b are associated with the' observed

frequencies as shown. The null-hypothesis is that the twin of a criminal is no more likely

to be convicted of crime if the twinning is monozygotic than if it is dizygotic. If the only

deviation from the hypothesis which we are prepared to consider is that monozygotic

twins behave more similarly than dizygotic, a single-tail test will be appropriate and we

shall ask whether a/A = 15/17 is significantly greater than b\B = 3/13.

Turning to the appropriate section of the table with A = 17, B = 13 and a = 15 we find

that the observed value of b = 3 is significant at the 0-5 % level, since it is less than 4, the

last entry in the row of bold figures.

The figure in small type following 6 = 4 indicates that for a contingency table with

marginal frequencies

4 = 17, - 5 = 1 3 , r = a + b= 15 + 4 = 19, A + B-a-b=ll,

the conditional probability of an arrangement within the table with 6 ^ 4 is 0-002 on the

null hypothesis of independence. The probability that b < 3 within the observed table

(having a + b = 18) is not recorded, but is < 0-002.

As far as possible, checks on the internal consistency of the table have been made as

well as comparisons with the more extensive tables for the special case A = B published

by Swaroop (1950).

REFERENCES

FINNBY, D. J. (1948). Biometrika, 35, 148.

FiSHEB, R. A. (1946). Statistical Methods for Research Workers, 10th ed. Edinburgh: Oliver and Boyd. SWABOOP, S. (1960). Indian Med. Res. Mem., no. 35.

A = 1 6 B = 16 '15 14 13 a 16 15 14 13 12 11 10 9 8 7 6 5 16 15 14 13 12 11 10 9 8 7 6 5 16 15 14 13 12 11 10 9 8 7 6 5 16 15 14 13 12 11 10 9 8 7 6 5 Significance tests in a Probability 0-05

11

-on

1 0 -041 8 "027 7 -033 6 "037 5 -038 4 -037 3 -033 2 -027 1 -019 1 -041 0 -022 1 1 -043 9 -033 8 -044 6 -023 5 024 4 -023 4 -049 3 -043 2 -033-1 -023

0 -on

0 -026 1 0 037 8 -023+ 7 -032 6 -033+ 5 -033+ 4 -033 3 -028 2 -021 2 -045-1 -030 0 -013 0 -031 9 -030 8 -047 6 -023 5 -023 4 -022 4 -048 3 -039 2 -029 1 -018 1 -038 0 -017 0 -037 0-025

11

-on

9 "019 7 -012 6 -015-5 -016 4 -016 3 -015-2 -01-015-2 1 -008 1 -019 0 -009 0 -022 1 0 -018 8 -014 7 -019 6 -023 5 -024 4 -023 3 020 2 O16 1 -010+ 1 -023

0 -on

—

9 -014 7 -oio-6 -013 5 -014 4 -014 3 -012 2 -009 2 -021 1 -013 0 -006 0 -013 —

8 -on

7 -019 6 -023 5 -023 4 -022 3 -018 2 -013 1 -008 1 -018 0 -007 0 -017 — 0-01 1 0 -009 8 -008 6 -005-5 -006 4 -006 3 -006 2 005-1 003 1 -008 0 -003 0 -009 — 9 -007 7 -003+ 6 -008 5 -009 4 -009 3 -008 2 -006 1 -004 0 -002 0 -004 — 8 -003+ 7 -oio-5 -003-4 -003+ 3 -003-2 -004 2 -009 1 -006 0 -002 0 -006 _ — 7 -004 6 -007 5 -oos 4 -008 3 -007 2 -003+ 1 -003 1 -008 0 -003 0 -007 — 0-005 9 -003 7 -003 6 -oos-4 -002 3 -002 2 -002 2 -005-1 -003 0 -ooi 0 -003 — 8 -002 6 -002 5 -003 4 -003 3 -O03 2 -002 1 -001 1 -004 0 402 0 -004 — 7 -002 6 -003 5 -005-3 -ooi 3 -005-2 004 1 -002 0 -ooi 0 -002 — 7 -004 5 -002 4 "003 3 -003 2 402 1 -001 1 -003 0 -ooi 0 "003 — 2 x 2 contingency table A = 1 6 B = 1 2 11 10 9 a 16 15 14 13 12 11 10 9 8 7 6 5 16 15 14 13 12 11 10 9 8 7 6 16 15 14 13 12 11 10 9 8 7 6 16 15 14 13 12 11 10 9 8 7 6 Probability 0-05 8 -024 7 -036 6 -040 5 -039 4 -034 3 -027 2 -019 2 -040 1 -024 1 -048 0 -021 0 -044 7 -019 6 -027 5 -027 4 -024 3 -019 3 -041 2 -028 1 -016 1 -033 0 -013 0 -027 7 -046 5 -018 4 -018 4 -042 3 -032 2 -021 2 -042 1 -023 1 -O4J-0 --O4J-017 0 -033-6 -037 5 040 4 -034 3 -023+ 2 -016 2 -033 1 -017 1 -034 0 -012 0 -024 0 -043+ 0-025 8 -024 6 -013 5 -013-4 -01-013-4 3 -012 2 -008 2 -019

1 -on

1 -024 0 -oio-0 --oio-021 — 7 -019 5 -009 4 -009 4 -024 3 -019 2 -013 1 -007 1 -016 0 -006 0 -013 — 6 -014 5 -018 4 -018 3 -014 2 -009 2 -021

1 -on

1 -023 0 -008 0 -017 — 5 -oio-4 -012 3 -oio-2 -007 2 -016 1 -008 1 -017 0 -006 0 -012 0 -024 — 0-01 7 -008 5 -004 4 -005-3 -004 2 -003 2 -008 1 -oos-0 --oos-0-oos-02 0 -004 0 -oio-— 6 -006 5 -009 4 -009 3 -008 2 -003+ 1 003 1 -007 0 -002 0 -006 — 5 404 4 -003+ 3 005-2 -003 2 -009 1 -005-0 --005-0-005-02 0 -004 0 -008 — 5 -oio-3 -00-oio-3 3 -oio-2 -007 1 -003 1 -008 0 -002 0 -006 — — 0-005 6 -002 5 -004 4 -005-3 -004 2 -003 1 -002 1 -003-0 --003-0-003-02 0 -004 — — 5 -002 4 -002 3 -002 2 -002 1 -001 1 -003 0 -ooi 0 -O02 — 5 -004 3 -ooi 3 -003-2 -003 1 -002 1 -005-0 --005-0-005-02 0 -004 — 4 -002 3 -003 2 -002 1 -001 1 -003 0 -ooi 0 -002 — — — —

The table shows:

(1) In bold type, for given A, B and a, the value of 6 (<<*) which is just significant at the probability levi quoted (single-tail test).

(2) In small type, for given A, B and r=a+b, the exact probability (if there is independence) that b is equal ( or lees than the integer shown in bold type.

Significance testa in a 2x2 contingency tmble (continued) A=16 B = 8 7 6 5 4 a 16 15 14 13 12 11 10 9 8 7 16 15 14 13 12 11 10 9 8 7 16 15 14 13 12 11 10 9 8 16 15 14 13 12 11 10 Q y 16 15 14 13 12 11 10 Probability 0-05 5 -028 4 -028 3 -021 3 -047 2 -028 1 -014 1 -027 0 -009 0 -017 0 "033 4 -020 3 -017 3 -045+ 2 -026 1 -012 1 -024 1 -04J-0 O14 0 -026 0 -047 3 -013 3 -046 2 023+ 1 Oil 1 -023 1 -043 0 -012 0 023 0 -040 3 448 2 -028 1 -Oil 1 -023+ 1 -047 0 012 0 423 0 -039 2 -032 1 -013 1 -032 0 -007 0 -014 0 -026 0 -043 0-025 4 -007 3 -007 3 -021 2 -013 1 406 1 -014 0 -004 0 -009 0 -017 4 -020

3 -on

2 on

1 -003-1 -0-003-12 1 -024 0 -007 0 -014 3 -013 2 009 1 -004 1 -Oil 1 023 0 -006 0 -012 0 -023 2 -008 1 -004 1 -Oil 0 -003 0 -006 0 -012 0 423 1 404 1 -013 0 -003 0 -007 0 -014 0-01 4 -007 3 -007 2 -005-1 -002 1 -006 0 -002 0 -004 0 -009 3 -004 2 -003 1 -002 1 -003-0 -ooi 0 -003 0 -007 2 -002 2 -009 1 004 0 ooi 0 -003 0 -006 2 -oos 1 404 0 -ooi 0 -003 0 -006 1 -004 0 -ooi 0 -003 0 -007 — 0-005 3 -ooi 2 -ooi 2 -003-1 -002 0 -ooi 0 -002 0 -004

z

3 -004 2 -003 1 -002 1 -ooj-0 -ooi 0 -003 — 2 402 1 -001 1 -004 0 -ooi 0 -003 1 -001 1 -004 0 -ooi 0 -003 — 1 404 1 -001 0 -003 —

A=16 B=3

2 A=17 B=17 16 15

a

16 15 14 13 12 16 15 14 17 16 15 14 13 12 11 10 9 8 7 6 5 17 16 15 14 13 12 11 10 9 8 7 6 5 17 16 15 14 13 12 11 10 9 8 7 6 5 0-05 1 -018 0 -004 0 -010+ 0 -021 0 -036 0 -007 0 -020 0 "039 12 -022 11 443 9 -029 8 -033+ 7 "040 6 -042 5 -042 4 040 3 -033+ 2 029 1 -020 1 -043 0 -022 1 2 -044 1 0 -033-9 046 7 -023+ 6 027 5 -027 4 -023+ 3 -022 3 -046 2 -ow 1 -024

0 -on

0 -026 11 "038 9 -027 8 -033+ 7 -040 6 -041 5 -039 4 433+ 3 -029 2 022 2 -046 1 -030 0 014 0 -031 Probability 0-025 1 -018 0 -004 0 -010+ 0 -021 — 0 -007 0 -020 12 -022 1 0 -020 8 -013 7 -016 6 -018 5 019 4 018 3 -016 2 -013 1 008 1 -020 0 -009 0 -022 11 -018 9 -oi J -8 -021

6 -on

5 -on

4 -on

3 -009 3 -022 2 -017 1 -on 1 -024

0 -on

— 10 415" 8 411 7 -013-6 -017 5 017 4 016 3 -013 2 -oio-2 -0-oio-2-oio-2 1 -014 0 -006 0 414 0-01 0 -ooi 0 -004 — — — 0 -007 — 11 409 9 -008 7 -003+ 6 -007 5 -007 4 -007 3 007 2 -003+ 1 -003 1 008 0 -004 0 -009 1 0 -007 8 -006 7 -009 5 -004 4 -004 3 -004 3 -009 2 -007 1 -004 0 -002 0 003-— — 9 "006 7 -004 6 -006 5 006 4 -006 3 -003+ 2 -004 2 -oio-1 -006 0 -002 0 -006 — 0-005 0 -ooi 0 -004 — — — — 10 -004 8 -003 6 402 5 -002 4 -003 3 -002 2 -002 1 -001 1 003 0 ooi 0 -004 — 9 -003 7 -002 6 -003 5 004 4 -004 3 404 2 403 1 402 1 404 0 402 0 403-— — 8 402 7 404 5 402 4 402 3 402 2 401 2 404 1 402 0 401 0 402 — —

A=17 B=14

13 12 11

a

17 16 15 14 13 12 11 10

9

8

7

6

5 17 16 15 14 13 12 11 10

9

8 7 6 5 17 16 15

14

13 12 11 10

9

8

7

6

17

16

15

14

S i g n i f i c a n c e t e s t a i n a 2 x 2

Probability

0O5

1 0 -032 8 -021 7 -026 6 -028 5 -027 4 024 4 -049 3 -040 2 -029 1 -018 1 -038 0 -017 0 -036 9 -026 8 O40 7 -045+ 6 -045+ 5 -042 4 033+ 3 -028 2 -019 2 040 1 -024 1 -047 0 -021 0 -043 8 -021 7 -030 6 -033 5 -030 4 -026 3 -02O 3 -041 2 -028 1 -016 1 -032 0 -012 0 -026 7 -016 6 -022 5 "022 4 -019

0-025

9 -012 8 -021

6

-oio-5 -on

4

-oio-4 -02-oio-4 3 -019 2 -014 1 -008 1 -018 0 -007 0 017 — 8 -009 7 -015+ 6 -018 5 -018 4 016 3 013 2 -009 2 -019 1 -Oil 1 -024

0

-oio-0 -oio-021 — 8 -021

6 -on

5 -012

4 on

3 -008 3 -020 2 -013 1 -007 1 -016 0 006 0 -012 — 7 -016 6 -022 5 -022 4 -019

0-01

8 -004 7 "008

6

-oio-4 00-oio-4

4

-oio-3 -008 2 -006 1 -003 1 -008 0 003 0 "007 — — 8 -009 6 -005+ 5 -006 4 -006 3 005+ 2 004 2 009 1 -005-C 002 0 -004

0

-oio-— — 7 007 5 -003 4 -004 3 -003 3 -008 2 -006 1 -003 1 -007 0 -002 0 006 — — 6 -003-5 -007 4 -007 3 -006 0-005 8 -004 6 403 5 -003 4 -004 3 -003 2 002 1 -001 1 -003

0 -ooi

0 -003 — — — 7 -003 5 -002 4 -002 3 -002 2 -ooi 2 -004 1 -002 1 -005-0 --005-0-005-02 0 -004 — — — 6 -002 5 -003 4 -004 3 -003 2 -002 1 -001 1 -003 0 -ooi 0 -002 — — — 6 -005-4 -002 3 -002 2 -ooi

contingency table (continued)

A=17 B=ll

10

9

8

a

13 12 11 10 9 8 7 6 17 16 15 14 13 12 11 10 9 8 7 6 17 16 15 14 13 12 11 10 9

8

7 17 16 15 14 13 12 11 10 9 8 7

0-05

4 -042 3 -031 2 -020 2 -040 1 -022 1 -042 0 -016 0 -033 7 -041 6 -047 5 -043 4 -034 3 -024 3 -049 2 -031 1 -016 1 -031

0 -on

0 -022 0 -042 6 -032 5 -034 4 -028 3 -020 3 -042 2 -025+ 2 -048 1 -024 1 -045-0 --045-016 0 -030 5 -024 4 -023 3 -017 3 -039 2 -022 2 -043 1 -020 1 -038 0 -012 0 -022 0 -040 Probability 0-025 3 -014 2 -009 2 -020

1 -on

1 -022 0 -008 0 016 6 -012 5 -015+ 4 -014 3 -010+ 3 -024 2 -015+ 1 -007 1 -016 0 -005+

0 -on

0 -022 — 5 -008

4

-oio-3 -008 3 -020 2 012 1 -006 1 -012 1 -024 0 -008 0 -016 — 5 -024 4 -023 3 -017

2

-oio-2 -0-oio-2-oio-2 1 -010" 1 -020 0 -006 0 -012 0 -022 — OOI 2 -004 2 -009 1 -005-0 -ooi 0 -OCM 0 -008 5 -003 4 -004 3 -004 2 -002 2 -007 1 -003 1 -007 0 402 0 -005+ — — — 5 -008

4

oio-3 -008 2 -005-1 -002 1 -006 0 -002 0 -004 0 -008 — — 4 -006 3 -006 2 -004

2

-oio-1 -004

1

oio-0 -oio-0oio-03 0 -006 — — — OKJ05 2 -004 1 -002 1

-003-0 -ooi

0 -004 5 003 4 -004 3 004 2 -002 1 -001 1 -003

0 -ooi

0 -003 — — — — 4 -002 3 -002

2 -ooi

2 -005-1 -002

0 -ooi

0 402 0 -004 — — —

3 -ooi

2 -ooi

2 -004 1 -002 1 -004

0 -ooi

0 -003

The table shows:

(1) In bold type, for given A, B and o, the value of 6 (<a) which ia just significant at the probability level

quoted (single-tail test).

(2) In small type, for given A, B and r = a+ b, the exact probability (if there is independence) that 6 ia equal to

or less than the integer shown in bold type.

Significance tests in a 2x2 contingency table (continued)

A=17 B=7

6 5 4 3 2

a

17 16 15 14 13 12 11 10 9

8

17 16 15 14 13 12 11 10 9

8

17 16 15 14 13 12 11

10

9

17 16 15 14 13 12 11 17 16 15 14 13 12 17 16 15 Probability 0-05 4 017 3 -014 3 -038 2 -021 2 -042 1 018 1 -034

0

-oio-0 --oio-019 0 -033

3 -on

3 -040 2 O21 2 -045+ 1 -018 1 435~ 0 -009 0 -017 0 -030 0 -030-3 -04-030-3 2 -024 1 -009 1 -021 1 -039 0 -oio-0 --oio-018 0 -030 0 449 2 -029 1 -Oil 1 -028 0 -006 0 -012 0 "021 0 435+ 1 -016 1 -046 0 -009 0 -018 0 -031 0 -049 0 -006 0 -018 0 "033+ 0-025 4 -017 3 -014 2 -009 2 021 1 -009 1 018 0 -005-0 -oio-0 --oio-019

3

-on

2 -008 2 -021 1 -009 1 -018 0 -003-0 --003-0-003-09 0 017 2 -006 2 -024 1 -009 1 -021 0 -005-0 -oio-0 --oio-018 — 1 -003

1 on

0 -003 0 -006 0 012 0 -021 — 1 -016 0 004 0 -009 0 -018 — — 0 -006 0 -018 — (M)l 3 -003 2 -003 2 -009 1 -004 1 -009 0 002 0 -005-0 -oio-2 -00-oio-2 2 -008 1 -003 1 -009 0 -002 0 -003-0 --003-0-003-09 2 -006 1 -003 1 -009 0 -002 0 -003-0 -oio-— 1 -003 0 -ooi 0 -003 0 -006 — — — 0 ooi 0 -004 0 -009 — — — 0 -006 —-0-005 3 -003 2 -003 1 -001 1 -004 0 -ooi 0 -002 0 -005-2 -00-005-2 1 -001 1 -003 0 -ooi 0 -002 0 -005-1 -00-005-1 1 -003 0 -ooi 0 -002 0 -003-— 1 -003 0 -ooi 0 -003 — — — — 0 -ooi 0 -004 — — — — — A = 1 8 B=18 17 16 15 a 18 17 16 15 14 13 12 11 10 9 8 7 6 5 18 17 16 15 14 13 12 11 10 9 8 7 6 5 18 17 16 15 14 13 12 11 10 9 8 7 6 5 18 17 16 15 14 13 Probability (H)5 13 -023 12 -044 10 -030 9 -038 8 -043 7 -046 6 -047 5 -046 4 -043 3 -038 2 -030 1 -020 1 -044 0 -023 13 445+ 11 436 10 -049 8 -028 7 -030 6 -031 5 -030 4 -028 3 -023 3 -047 2 -037 1

-023-0 -on

0 -026 12 -039 10 -029 9 -038 8 -043 7 -046 6 -043+ 5 -042 4 -037 3 431 2 -023 2 -046 1 -030 0 -014 0 -031 11 -033 9 -023 8 -029 7 "031 6 -031 5 -029 0-025 13 -023 11 420 9 -014 8 -018 7 "020 6 -022 5 -022 4 -020 3 -018 2 -014 1 -009 1 -020

0

-oio-0 --oio-023 12 -019 10 -016 9 -023 7 -012 6 413 5 413 4 412 3 410+ 3 423 2 418 1 411 1 425-0 411 — 11 416 9 412 8 417 7 419 6 420 5 420 4 418 3 415-2 411 2 423 1 414 0 406 0 414 10 413 9 423 7 412 6 413 5 413 4 411 OOI 12 410" 10 409 8 406 7 408 6 409 5 409 4 409 3 408 2 006 1 404 1 409 0 404 0 410-— 11 408 9 407 8 410-6 403-5 403+ 4 405-3 404 2 403 2 408 1 405-0 4405-02 0 405-— — 10 406 8 403" 7 407 6 408 5 408 4 407 3 406 2 404 1 403 1 406 0 402 0 406 9 405-8 409 6 404 5 405-4 405-40405-4 3 404 0-005 11 404 9 404 7 002 6 403 5 403 4 403 3 403 2 402 1 401 1 404 0 401 0 404 — — 1 0 OOJ 8 402 7 404 6 403-4 403-402 4 403-3 404 2 403 1 402 1 405-0 4405-02 0 403-— — 9 402 8 403" 6 402 5 403 4 403 3 402 2 402 2 004 1 403 0 401 0 402 9 405-7 403 6 404 5 405-4 405-40405-4 3 404

A = 1 8 B = 1 5 14 13 12 a 12 11 10 9 8 7 6 5 18 17 16 15 14 13 12 11 10 9 8 7 6 5 18 17 16 15 14 13 12 11 10

9

8

7

6

18

17

16 15 14 13 12 11 Significance tests in a 2 x 2 Probability 0-05 4 -023+ 3 -020 3 -041 2 -030 1 -018 1 -0J8 0 -017 0 -036 10 -028 9 -043 8 -050-6 -022 6 -049 5 -044 4 -037 3 021 2 -020 2 O39 1 -024 1 -047 0 -020 0 -043 9 023 8 034 7 037 6 -036 5 -032 4 -027 3 -020 3 040 2 -027 1 -015+ 1 031 0 -012 0 -023+ 8 -018 7 -026 6 -027 5 -024 4 -020 4 -042 3 -030 2 -019 0-025 3 -009 3 -020 2 -014 1 -008 1 -018 0 -007 0 -017 9

-oio-8 -on

7 -021 6 -022 5 -020

4 -on

3 -013 2 -009 2 -020

1 -on

1 -024 0 -009 0 -020 9 -023 7 -012 6 -014 5 -014 4 -012 3 009 3 -020 2 -013 1 -007 1 -013+ 0 -006 0 -012 8 -018 6 -009 5 -009 5 -024 4 -020 3 -014 2 -009 2 -019 0-01 3 -009 2 -006 1 -004 1 -008 0 -003 0 -007 9 -oio-7 -006 6 -008 5 -008 4 -007 3 -ooe 2 -004 2 -009 1 -O03-0 -O-O03-02 0 -004 0 -009 8 -008 6 -004 5 -003-4 -00-003-4 3 -004 3 -009 2 -006 1 -003 1 -007 0 -002 0 -006 7 -006 6 -009 5 -009 4 -008 3 -006 2 -004 2 -009 1 -003-0-005 2 -003 1 -001 1 -004 0 -ooi 0 -003 8 -003 6 -002 5 -003 4 -003 3 -002 2 -ooi 2 -004 1 -002 1 -003-0 --003-0-003-02 0 -004 7 -002 6 -004 5 -003-4 -00-003-4 3 -004 2 -002 1 -001 1 -003 0 -ooi 0 -002 6 -002 5 -003 4 -003 3 -002 2 -ooi 2 -004 1 -002 1

-003-contingency table (continued)

A = 1 8 B = 1 2 11 10 9

a

10 9 8 7 6 18 17 16 15 14 13 12 11 10

9

8

7

6

18

17

16 15 14 13 12 11 10 9 8 7 6 18 17 16 15 14 13 12 11 10 9 8 7 Probability 0K)5 2 -038 1 -021 1 -040 0 -016 0 -031 8 043+ 6 -018 5 018 5 -043 4 -033 3 -023 3 -046 2 -029 1 -015-1 -029 0 -010+ 0 -020 0 -039 7 -037 6 «41 5 -036 4 -028 3 "019 3 -039 2 -023 2 -043 1 -022 1 -040 0 -014 0 027 0 -049 6 -029 5 -030 4 023 3 -016 3 O34 2 -019 2 -037 1 -018 1 -033 0 -010+ 0 -020 0 -036 0-025 1 -010+ 1 -021 0 -007 0 -016 7 -014 6 -018 5 -018 4

-015-3 -on

3 -023 2 -014 1 -007 1 -013-0 --013-0-013-03- -003-0 --003-01-003-0+ 0 -020 6 -010+ 5 -013

4 -on

3 -008 3 -019

2 -on

2 -023

1 -on

1 -022 0 -007 0 -014 — — 5 -007 4 -008 4 -023 3 -016 2 -009 2 019 1 -009 1 -018 0 -003+ 0 -010+ 0 -020 0-01 0 -ooi 0 -003 0 -007 — — 6 -004 5 -006 4 -003+ 3 -004 2 -003 2 -007 1 -003 1 -007 0 -002 0 -003-— — — 5 -O03 4 -003 3 -O03 3 -008 2 -003-1 -002 1 -003+ 0 ooi 0 -003 0 -007 — — — 5 007 4 -008 3 -006 2 -004 2 -009 1 -004 1 -009 0 002 0 -005+ — — 1 — 0-005 0 ooi 0 -O03 — — — 6 -004 4 -ooi 3 ooi 3 -004 . 2 -003 1 -001 1 -003 0 ooi 0 -002 0 -003-— — — 5 -003 4 -003 3 -003 2 -002 2 -003-1 -002 0 -ooi 0 -ooi 0 -003 — — — — 4 -002 3 -002 2 ooi 2 -O04 1 002 1 004 0 -ooi 0 "002 — — —

The table shows:

(1) In bold type, for given A, B and a, the value of b ( < o ) which is just significant at the probability level quoted (single-tail test).

(2) In small type, for given A, B and r = a + b, the exact probability (if there is independence) that b is equal to or lees than the integer shown in bold type.

Significance tests in a 2x2 contingency table (continued)

A=18

B = 8

7 6 5 4 a 18 17 16 15 14 13 12 11 10 9

8

7

18 17 16 15 14 13 12 11 10 9 8 18 17 16 15 14 13 12 11 10 9 18 17 16 15 14 13 12 11 10 18 17 16 15 14

0-05

5 4

3

3

2 2 1 1 1 0

0

o

4 3 3 2 2 1 1 1 0 0 0 3 3 2 2

1

1

1

0 0 0 3 2 2 1 1 0 0 0 0 2 1 1 1 0 •022 •020 •014 •032 •017 •034 •015+ •028 -049 •016 •028 -043 •015+ •012 -032 •017 •034 •014 •027 •046 •013 •024 •040 •oio-•035+ •018 •038 • 0 1 5 -•028 •048 •013 •022 •037 •040 •021 -048 •017 •033 •007 -014 •024 •038 •026 •oio-•024 •046 •010" Probability 0-025

5

4

3

2 2 1 1 0 0 0 4 3

2

2 1 1 0 0 0 0 3 2 2 1 1 0 0 0 0 — 2 2 1 1 0 0 0 0 — 1 1 1 0 0 •022 •020 -014 •008 •017 •007 •015+ -004 •008 •016 •015+ •012 •007 •017 •007 •014 •004 -007 •013 •024 •oio-•006 •018 •007 •015-•003 •007 •013 •022 •006 021 •008 •017 •004 •007 •014 •024 003 •oio-•024 •005- •oio-0-01 4 -005-3 -004 2 -003 2 -008 1 -003 1 -007 0 -002 0 -004 0 -008 — 3 003 2 "002 2 -007 1 -003 1 -007 0 -002 0 004 0 -007 — — — 3 -oio-2 -006 1 -003 1 -007 0 002 0 -003 0 -007 — — — 2 -006 1 -003 1 -008 0 -002 0 '004 0 -007 — — — 1 -003 1 -oio-0 '-oio-0-oio-02 0 -005-0 -oio-0-005 4 3 2 1 1 0 0 0 — 3 2

1

1 0 0 0 — — — 2 1 1 0 0 0 — — — — 1 1 0 0 0 — — — — 1 0 0 0 •005-•004 •003 •001 •003 •001 •002 •004 •003 •002 •001 •003 •001 •002 •004 •001 •001 003 •001 •002 •003 •001 003 •001 •002 •004 003 •001 •002 • 0 0 5 "

A=18

B = 4

3 2

A=19 B=19

18 17 a 13 12 11 18 17 16-15 14 13 18 17 16 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 19 18 17 16 005 0 0 0 1 1 0 0 0 0 0 0 0 14 13 11 10 9 8 6 5 5 4 3 2 1 1 0 14 12 10 9 8 7 6 5 4 3 3 2 1 0 0 13 11 10 9 •017 •029 •045+ •014 441 -008 •015+ •026 •042 •005+ -016 •032 •023 •045-•031 •039 •046 •050-•025+ 024 • 0 5 0 " -046 •039 •031 K)21 •045" •023 •046 •037 •024 030 •033 •035+ •035" •033 •030 •025" •049 •038 •025+ -012 •027 •040 •030 •040 •047 Probability 0-025 0 — — 1 0 0 0 — — 0 0 — 14 12 10 9 8 7 5 5 4 3 2 1 1 0 0 13 11 10 8 7 6 5 4 3 3 2 1 0 0 12 10 9 8 •017 •014 •003 •008 •015+ •005+ •016 •023 021 •015-019 •022 •024 •Oil •024 •022 •019 •015-•009 •021 •oio-•023 •020 •017 •024 014 •015+ •016 •015+ 014 •Oil •025-•019 •012 • 0 0 5 -•012 •016 013 •018 •022 001 0 -ooi 0 -003 0 008 — — — 0 -005+ — — 13 oio-1 oio-1 -009 9 -006 8 009 6 -004 5 -004 4 -004 3 -003 3 009 2 -006 1 -004 1 -009 0 -O04 0 oio-1 2 -008 10 007 8 004 7 -006 6 006 5 -006 4 006 3 -005-2 -004 2 -008 1 005+ 0 002 0 -ooj-— 11 -006 9 -005+ 8 -008 7 -009 0O05 0 0 — — — — — — — 12 10 8 7 6 5 4 3 2 1 1 0 0 11 9 8 6

5

4 3 3 2 1 0 0 0 — 10 8 7 6 •001 •003 •004 •004 •003 •003 •004 •004 004 -003 •003 •002 •004 •002 •004 •003 •003 •004 •002 •002 •002 002 • 0 0 5 -•004 •002 001 •002 • 0 0 5 " 002 -002 -003 •003 Biometrika 40

Significance tests in a 2x2 contingency table (continued)

A=19 B=17

16 15 14 a 15 14 13 12 11 10 9 8

7

6

5

19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 19 18 17

Probability

0-05

8 -050-6 -023 6 049 5 -043-4 -039 3 -032 2 -024 2 047 1 -031 0 -014 0 -031 1 2 -035-1 0 -024 9 -031 8 -035-7 036 6 034 5 -031 4 -027 3 021 3 -042 2 -030 1 -018 1 -037 0 O17 0 -036 11 029 1 0 -046 8 -023 7 023-6 -024 5 022 5 -045+ 4 -037 3 -029 2 -020 2 -039 1 -023 1 -046 0 -020 0 "042 10 -024 9 -037 8 042

0025

7 -023 6 -023 5 -022 4 -019 3 015+

2 -on

2 -024 1 -015-0 --015-0-015-06 0 -014 — 1 1 -013 1 0 -024 8 -013 7 -013+ 6 013+ 5 -014 4 013 3 -oio-3 -021 2 -015-1 009 1 -018 0 -007 0 -017 —

10 -on

9 -019 8 -023 7 -023-6 -024 5 -022 4 -018 3 '014 2 009 2 -020

1 -on

1 -023 0 -009 0 -020 . 1 0 -024 8 -014 7 -017 0 0 1 6 5 -oio-4 -008 3 -007 2 003" 1 -003 1 -007 0 -002 0 -006 — 10 -003-9 -oio-7 -005+ 6 006 5 006 4 003+ 3 004 3 oio-2 -007 1 -004 1 009 0 003 0 007 — — 9 -004 8 -007 7 -009 6 -oio-5 -009 4 008 3 -006 2 004 2 009 1 -003+ 0 002 0 -004 0 009 9 -008 7 -003-6 00-003-6 0-005 5 -004 4 003 3 -003 2 -002 2 -005-1 -003 0 ooi 0 -002 — 10 003-8 004 6 002 5 -002 4 -002 3 -002 3 004 2 -003 1 -002 1 004 0 ooi 0 003 — — — 9 004 7 002 6 -003 5 -003 4 -003 3 -002 2 -002 2 -004 1 -002 0 -ooi 0 -002 0 '004 — 8 003 7 003-5 -002.

A=19 B=14

13 12 11 a 16 15 14 13 12 11 10 9

8

7

6

5

19 18 17 16 15 14 13 12 11 10 9 8 7 6 19 18 17 16 15 14 13 12 11 10 9 8 7

6

19

18

17

16

Probability 0-05 7 -042 6 039 5 -034 4 -027 3 020 3 -040 2 -027 1 -013-1 -030 0 012 0 024 0 -049 9 -020 8 029 7 031 6 029 5 025+ 4 -020 4 041 3 029 2 019 2 036 1 020 1 038 0 015-0 015-03015-0 9 049 7 -022 6 022 5 019 5 042 4 -032 3 -023 3 -043 2 -027 2 -050-1 -027 1 050-0 050-019 0 -037 8 041 7 -047 6 -043 5 -035+ 0-025 6 -017 5 015+ 4 013 3 -009 3 -020 2 -013 1 007 1 -015-0 --015-0-015-05+ 0 012 0 024 — 9 -020 7 010+

6 -on

5 on

4 -009 4 -020 3 015" 2 009 2 019 1 oio-1 -020 0 -O07 0 -015-— 8 -016 7 -022 6 022 5 -019 4 -015+

3 -on

3 023 2 -014 1 007 1 -014 0 005" 0 -oio-0 --oio-019 — 7 012 6 -016 5 015-4 -012 0 0 1 5 006 4 005+ 3 004 3 -009 2 -006 1 -003 1 007 0 002 0 003+ — — 8 006 6 -003 5 004 4 003 4 -009 3 006 2 004 2 009 1 005-1 oio-0 -oio-0oio-03 0 -007 — — 7 -005-6 -007 5 007 4 006 3 -004 2 -003 2 -006 1 003 1 -007 0 -002 0 003-0 -oio-6 003 5 -004 4 004 3 -003

0005

4 -002 3 -ooi 3 004 2 -003 1 -001 1 003 0 -ooi 0 -002 — — — 7 002 6 -003 5 004 4 003 3 003 2 002 2 -004 1 -002 1 003" 0 ooi 0 003 — — — 7 003-5 002 4 -002 3 -002 3 004 2 003 1 001 1 -003 0 ooi 0 002 0 005-— — — 6 003 5 -004 4 -004 3 003

The table shows:

(1) In bold type, for given A, B and a, the value of 6 (<a) which is just significant at the probability level

quoted (single-tail test).

(2) In small type, for given A, B and r = a+b, the exact probability (if there is independence) that b is equal to

or less than the integer shown in bold type.

Significance tests in a 2 x 2 oontingeney bible (continued) A = 1 9 B = l l 10 9 8 a 15 14 13 12 11 10 9 8 7

6

19 18 17 16 15 14 13 12 11 10 9 8 7 19 18 17 16 15 14 13 12 11 10

9

8 7 19 18 17 16 15 14 13 12 11 10 9 8 Probability 0-05 4 -027 3 -018 3 -035+ 2 -021 2 -040 1 -020 1 -037 0 -013 0 -025-0 --025-046 7 "033 6 -036 5 -030 4 -022 4 -047 3 -030 2 -017 2 -033 1 -016 1 -029 0 -009 0 -018 0 -032 6 -026 5 -026 4 -020 4 -044 3 -028 2 -015-2 -0-015-29 1 -013 1 -024 1 -042 0 -013 0 024 0 -043 5 -019 4 -017 4 -044 3 -027 2 -013 2 -027 2 -049 1 -021 1 033 0 -on 0 -020 0 -034 0-025 3 -008 3 -018 2 -010+ 2 -021 1 -010+ 1 -020 0 -006 0 -013 0 -023-— 6 -009

5 -on

4 -009 4 -022 3 -013-2 -008 2 -017 1 -008 1 -016 0 -003-0 O-003-09 0 -018 — 5 -006 4 -007 4 -020 3 -013 2 -007 2 -013-1 -006 1 -013 1 024 0 -007 0 -013 0 -024 5 -019 4 -017

3 -on

2 -006 2 -013 1 -006 1 -Oil 1 -021 0 -006

0 -on

0 -020 — 0-01 3 -008 2 -003-1 -002 1 -003-0 -ooi 0 -003 0 -006 — — — 6 -009 4 -003 4 -009 3 -006 2 -004 2 -008 1 -004 1 -008 0 -002 0 "003-0 -"003-0"003-09 — — 5 -006 4 -007 3 -003-2 -003 2 -007 1 -003 1 -006 0 -002 0 -004 0 -007 4 -004 3 -004 2 -002 2 -006 1 -002 1 -006 0 -ooi 0 -003 0 -006 — 0-005 2 -002 2 -005-1 -002 1 -005-0 -ooi 0 -003 — — — 5 -002 4 -003 3 -002 2 -oot 2 -004 1 -002 1 -004 0 -ooi 0 -002 0 -003-— 4 -ooi 3 -ooi 3 -003-2 -003 1 -001 1 -003 0 -ooi 0 -002 0 -004 4 -004 3 -004 2 002 1 -001 1 -002 0 -ooi 0 -ooi 0 -003 — A = 1 9 B = 7 6

5

4 3 a 19 18 17 16 15 14 13 12 11 10 9 8 19 18 17 16 15 14 13 12 11 10 9 19 18 17 16 15 14 13 12 11 10 19 18 17 16 15 14 13 12 19 18 17 16 15 14 Probability 0-05 4 -013 4 -047 3 -028 2 -014 2 -028

1 on

1 -021 1 -037 0 -oio-0 --oio-017 0 -030 0 -048 4 -030-3 -0-030-31 2 -015+ 2 -032 1 -012 1 -023 1 -039 0 -oio-0 --oio-017 0 -028 0 -043+ 3 -036 2 -018 2 -042 1 -014 1 028 1 -047

0 -on

0 -019 0 -030 0 -047 2 -024 1 -009 1 -021 1 -040 0 003 0 -014 0 -024 0 -037 1 -013 1 -038 0 -006 0 -013 0 -023 0 -036 0O25 4 -013 3 -010+ 2 -006 2 -014 1 -003+ 1 -Oil 1 -021 0 -003+ 0 -oio-0 --oio-017 — 3 -009 2 005+ 2 "015+ 1 -006 1 -012 1 -023 0 -003+ 0 -oio-0 --oio-017 — 2 -005-2 -018 1 -006 1 -014 0 -003 0 -006

0 -on

0 -019 2 -024 1 009 1 -021 0 -004 0 -008 0 -014 0 -024 — 1 -013 0 -003 0 -006 0 -013 0 -023 — 0-01 3 -002 2 002 2 -006 1 -002 1 -005+ 0 ooi 0 -003 0 -003+ 0 -oio-— — 3 -009 2 -003+ 1 -002 1 -006 0 ooi 0 -003 0 -00S+ 0 -oio-— — 2 -005-1 -002 1 -006 0 -ooi 0 -003 0 -006 — 1 -002 1 -009 0 -002 0 -004 0 -008 — — — 0 ooi 0 "003 0 -006 — — 0-005 3 -002 2 -002 1 -001 1 002 0 -ooi 0 -oot 0 -003 — — 2 -ooi 1 -001 1 -002 0 -ooo 0 ooi 0 -003 — — — 2 003-1 -002 0 -ooo 0 -ooi 0 -003 — — — 1 -002 0 -ooi 0 -002 0 -004 — — — — 0 -ooi 0 -003 — — — 6-a

A = 1 9 B = 2 A = 2 0 B = 20 19 18 a 19 18 17 16 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 20 19 18 ; 17 16 Significance testa in a 2 x 2 Probability 0-05 0 -005-0 -005-014 0 -029 C-048 15 -024

14

-m

12 -032 11 -041 10 -048 8 -027 7 028 6 -028

5

on

4 -024 4 -048 3 -041 2 -032 1 -022 1 -046 0 -024 15 -047 13 -039 11 -026 10 -032 9 -036 8 038 7 039 6 038 5 -035+ 4 -031 3 -026 2 -019 2 -039 1 -026 0 -012 0 -027 14 "041 12 -032 11 -043 10 -050-8 -026 0-025 0 -005-0 -005-014 15 -024 13 -022 11 415+ 1 0 -020 9 -024 7 -012 6 -013 5 -012

4

-on

4 -024 3 -020 2 O15+ 1 -oio-1 -022 0 -010+ 0 -024 14 020 12 -018 10 -012 9 015-8 -017 7 -018 6 -018 5 -017 4 015+ 3 012 2 009 2 -019 1 -012 0 405+ 0 -012 —

13

-on

11 -014 10 -020 9 -024

7

-on

0-01 0 -005-13 404 12 -oio-10 -007 9 -009 7 -005-6 -005+ 5 -005+ 4 -005-3 -004 3 -009 2 -007 1 -004 1 -oio-0 --oio-0-oio-04 — — 13 -008 11 408 9 -005-8 006 7 007 6 -008 5 -007 4 -007 3 -005+ 2 -004 2 -009 1 405+ 0 -002 0 -005+ — — 12 007 10 006 9 -008 7 -004 6 -005-0-005 0 -005-13 404 11 404 9 -003 8 -004 7 -003-5 -002 4 -002 4 -005-3 -004 2 -003 1 -002 1 -004 0 -002 0 -004 — — 12 -003 10 003 9 405" 7 -002 6 -003 5 -003 4 -003 3 -002 2 -002 2 -004 1 -002 0 -ooi 0 -002 — — — 11 403 9 -002 8 -003 7 -004 6 405"

contingency bMe (continued)

A = 20 B=18 17 16 a 15 14 13 12 11 10 9 8 7 6 5 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 Probability 0O5 7 -027 6 -026 5 -024 5 -047 4 041 3 -033 2 -024 2 -048 1 -031 0 -014 0 031 13 -036 11 026 10 -034 9 -038 8 -040 7 -039 6 -037 5 -033 4 -028 3 -022 3 -042 2 -031 1 -019 1 -037 0 -017 0 -036 12 -031 11 049 9 -026 8 -028 7 -028 6 -026 5 -023 5 -046 4 -038 3 -029 2 -020 2 -039 1 -023 1 445+ 0 -020 0 -041 0-025 6 412 5 4ii 5 -024 4 -020 3 -016 2 -012 2 -024 1 -015-0 --015-0-015-06 0 -014 — 12 -014

10

-on

9 -015-8 -017 7 -018 6 -017 5 -016 4 413 3 -010+ 3 422 2 -015+ 1 409 1 419 0 408 0 417 — 11 412 10 021 8 411 7 412 6 412 5 Oil 5 023 4 419 3 414 2 410-2 4410-20 1 411 1 423 0 409 0 420 0-01 5 404 4 404 4 409 3 407 2 405+ 1 403 1 407 0 003 0 406 — — 11 405+ 9 404 8 406 7 407 6 407 5 407 4 406 3 405" 2 403 2 407 1 404 1 409 0 403 0 408 — — 10 004 9 008 7 404 6 404 5 404 4 004 4 409 3 407 2 404 2 410-1 405+ 0 402 0 404 0 409 — 0-005 5 404 4 404 3 403 2 402 1 401 1 003 0 401 0 403 — — — 10 402 9 404 7 002 6 403 5 403 4 402 3 402 3 405-2 403 1 402 1 404 0 401 0 403 — — — 10 404 8 403 7 404 6 404 5 404 4 404 3 003 2 402 2 404 1 402 0 401 0 402 0 404 — —

The table shows:

(1) In bold type, for given A, B and a, the value of b ( < o ) which is just significant at the probability leve quoted (single-tail test).

(2) In small type, for given A, B and r = o + 6 , the exaot probability (if there is independence) that 6 is equal U. or less than the integer shown in bold type.

Significance tests in a 2x2 contingency table (continued) A = 2 0 B=15 14 13 12 a 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 20 19 Probability 0-05 11 -026 10 -040 9 -046 8 -047 7 -045-6 040 5 -034 4 -028 3 -020 3 -039 2 -026 2 -049 1 -029 0 "012 0 -024 0 44S 10 -022 9 -032 8 -035+ 7 -035-6 -031 5 -026 4 420 4 -040 3 -029

2 -on

2 -035+ 1 -019 1 -037 0 -014 0 -029

9 -on

8 023-7 -026 6 024 5 020 5 -041 4 -031 3 -022 3 -041 2 -026 2 -047 1 -026 1 -047 0 -018 0 -033-9 044 7 -019 0-025 10 -009 9 -016 8 -019 7 -020 6 -019 5 417 4 413 3 -oio-3 020 2 "013 1 -007 1 -015-0 4-015-05+ 0 -012 0 -024 10 422 8 -012 7 -014 6 -013 5 -012 4 -009 4 -020 3 -015-2 -009 2 -018 1 -oio-1 -0-oio-19 0 -007 0 414 9 417 8 423" 6 409 6 024 5 420 4 415+ 3 on 3 422 2 413 1 407 1 413 0 404 0 409 0 418 8 414 7 419 0-01 10 409 8 406 7 407 6 408 5 407 4 406 3 404 3 410-2 406 1 403 1 407 0 402 0 405+ 9 407 7 404 6 5 405- 405-4 405-40405-4 4 409 3 407 2 404 2 409 1 405-1 4405-10- 410-0 4410-03 0 407 8 405+ 7 408 6 409 5 408 4 407 3 405-2 403 2 406 1 403 1 007 0 402 0 404 0 409 7 404 6 406 0-005 9 403 7 402 6 402 5 402 4 402 3 402 3 404 2 403 1 401 1 403 0 401 0 402 — , 8 402 7 404 6 5 405- 405-4 405-40405-4 3 403 2 402 2 404 1 402 1 405-0 4405-01 0 403 7 002 6 403 5 403 4 402 3 002 3 005-2 403 1 401 1 403 0 401 0 402 0 404 7 404 5 402 A = 2 0 B=12 11 10 9 a 18 17 16 15 14 13 12 11 10 9 8 7 6 20 19 18 17 16 15 14 13 12 11 10 9 8 7 20 19 18 17 16 15 14 13 12 11 10 9 8 7 20 19 18 17 16 15 14 Probability ' 0-05 6 418 6 443 5 434 4 425+ 4 449 3 433 2 420 2 436 1 418 1 434 ' 0 412 0 423 0 443 8 437 7 442 6 437 5 429 4 421 4 442 3 428 2 416 2 429 1 414 1 426 1 446 0 416 0 429 7 430 6 431 5 426 4 418 4 439 3 424 3 445+ 2 423+ 2 445-1 42445-1 1 437 0 412 0 422 0 438 6 423 5 422 4 416 4 437 3 022 3 443 2 423 0-025 6 418 5 416 4 412 3 408 3 417 2 410-2 4410-20 1 409 1 418 0 406 0 412 0 423 — 7 410+ 6 413 5 412 4 409 4 421 3 414 2 408 2 416 1 407 1 414 0 404 0 408 0 416 — 6 408 5 409 4 407 4 418 3 412 3 424 2 413 1 406 1 411 1 421 0 406 0 412 0 422 — 6 423 5 422 4 416 3 410+ 3 422 2 412 2 423 0-01 5 406 4 403-3 -00403-3 3 408 2 -003-2 410-1 405-1 409 0 403 0 406 — — — 6 403 5 404 4 403 4 -009 3 406 2 403 2 408 1 403 1 407 0 402 0 404 0 408 — — 6 408 5 409 4 407 3 005-2 403 2 406 1 403 1 -006 0 401 0 403 0 406 — — — 5 403+ 4 405+ 3 404 2 402 2 403+ 1 402 1 403" 0-005 4 402 4 405-3 40405-3 2 402 2 405-1 402 1 405" 0 401 0 003 — — — — 6 403 5 404 4 403 3 002 2 401 2 403 1 401 1 403 0 401 0 402 0 004 — — — 5 402 4 402 3 402 3 405-2 003 1 401 1 403 0 401 0 401 0 003 — — — — 4 401 3 401 3 404 2 402 1 401 1 402 1 405"

Significance tests in a 2x2 contingency table (continued)

A=20 B = 9

8 7 6 a 13 12 11 10 9 8 7 20 19 18 17 16 15 14 13 12 11 10 9 8 20 19 18 17 16 15 14 13 12 11 10 9 20 19 18 17 16 15

0-05

2 1 1 0 0 0 0 5 4 4 3 3 2 2 1

1

1

0 0 0 4 4 3

3

2

2

1

1

1

0

0

0

4

3

2

2

1

1

•041 •018 •032 •009 •017 •029 •OJO-•017 •015-•033 •022 •044 •022 -040 •016 •029 •048 •014 •024 •041 •012 •042 •024 •050-•023 •043 •016 •029 •048 •013 •022 •036 •046 •028 •013 •028 •oio--018 Probability 0-025 1 -009 1 -018 0 -005-0 --005-0-005-09 0 -017 5 -017 4 -015-3 -009 3 -022

2 -on

2 -022 1 -009 1 -016 0 -004 0 -008 0 «14 0 -024 4 -012 3 -009 3 -024

2 -on

2 -023 1 -O09 1 -016 0 -004 0 007 0 -013 0 -022 — 3 -008 2 -O05-2 -013 1 -O04 1 -oio-1 0-oio-18 0-01

1

0 0 0 4 3 3

2

1

1 1 0 0 0 3 3 2 1 1 1 0 0 0 — 3 2 1 1

1

0

•009 -002 •005-•009 •003 •003 -009 •005-•002 •004 •009 •002 -004 •008 •002 •009 •003-•002 •004 009 •002 •004 •007 •008 • o o j -•002 •004 •oio-•002 0-005 0 0 0 4 3 2 2 1 1 0 0 0 3 2 2 1 1 0 0 0 — 2 2 1 1 0 0 •001 -002 • 0 0 3 -•003 •003 •002 •005-•002 •004 •001 002 •004 •002 •001 • 0 0 5 -•002 •004 •001 •002 -004 •001 •005-•002 •004 •001 •002

A=20 B = 6

5 4 3 2 1 a 14 13 12 11 10 20 19 18 17 16 15 14 13 12 11 20 19 18 17 16 15 14 13 12 20 19 18 17 16 15 14 20 19 18 17 20

-0-05

1 0 0 0 0 3 2 2 1 1 1 0 0 0 0 2 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 •032 •007 •013 •022 •035-•033 •016 •038 •012 •023 •040 •009 •015--024 •038 •022 -008 •018 •035+ •007 •012 •020 •031 •047 •012 •034 •006 •Oil •020 •032 •047 •004 •013 •026 •043 •048 Probability 0-025 0 0 0 0 2 2 1 1 1 0 0 0 0 2 1 1 0 0 0 0 1 0 0 0 0 — 0 0 — •004 •007 •013 •022 •004 •016 •005+ -012 •023 •005" •009 •013-•024 •022 •008 •018 •003 •007 •012 •020 •012 -002 •006 •on •020 •004 -013 0-01 0 0 2 1 1 0 0 0 0 1 1 0 0 0 — — 0 0 0 —

0

—

•004 •007 •004 •002 •005+ •001 •002 •005-•009 •002 •008 001 •003 •007 •001 •002 •006 •004 0-005 0 — — — — 2 1 0 0 0 0 — — — — 1 0 0 0 — — — — — 0 0 — — — — — 0 — — — — •004 •004 •002 •000 -001 002 •005--002 •000 •001 •003 •001 •002 •004

The table shows:

(1) In bold type, for given A, B and a, the value of 6 ( < a ) which is just mgniJWnt. at the probability level quoted (single-tail test).

(2) In small type, for given A, B and r=a + b, the exact probability (if there is independence) that 6 is equal tc or less than the integer shown in bold type.